An adaptive homotopy method for permeability estimation of a nonlinear diffusion equation

Abstract

We investigate the problem of estimating the permeability function in a nonlinear diffusion equation, which plays an important role in promoting the permeability estimation within multiphase porous media flow. The forward problem is discretized using finite-difference methods and the parameter estimation is formulated as a least-square minimization problem with a regularization term. To overcome the weakness of the local convergence of traditional methods, a homotopy method is applied to solve this inverse problem. By introducing the Tikhonov regularization and adaptively choosing the homotopy parameters, a new and globally convergent algorithm is constructed. Finally, many numerical simulations are presented to show the global convergence, computational efficiency and anti-noise ability of the proposed algorithm.

A Homotopy Method for the Constrained Inverse Problem in the Multiphase Porous Media Flow

“…In previous works [44,45], we have verified the effectiveness of the homotopy and multigrid-homotopy methods for the inverse problem of the nonlinear diffusion equation:…”

“…which is an intermediate step of permeability identification in multiphase porous media flow. Different from [44,45], this paper not only considers the permeability identification based on the nonlinear convection-diffusion Equation ( 1), which can more accurately describe the multiphase flow process in porous media than the nonlinear diffusion Equation ( 6), but also introduces a well-log constraint to this inverse problem. The resulted constrained inverse problem can be transformed into a nonlinear constrained optimization problem.…”

A Regularization Homotopy Strategy for the Constrained Parameter Inversion of Partial Differential Equations

“…Shidfar et al [13] used a weighted homotopy analysis method to solve the inverse problem of identifying an unknown source term in a parabolic equation. Zhao et al [14] developed an adaptive homotopy method for the parameter identification inverse problem of a nonlinear diffusion equation. Hu et al [15] constructed a homotopy approach to improve the PEM identification of ARMAX models.…”